The Meaning of Solving a Polynomial Equation

Suppose we have two functions [latex]f(x)=x^2-2x-10[/latex] and [latex]g(x)=5[/latex]. Using Desmos to graph these functions we get the graph in figure 1.

Figure 1. The intersection of two functions

The horizontal line representing [latex]g(x)=5[/latex] intersects the parabola representing [latex]f(x)=x^2-2x-10[/latex] in two points. The two graphs intersect at the points (–3, 5) and (5, 5). These are the points where [latex]f(x)=g(x)[/latex] or, in this case, where [latex]x^2-2x-10=5[/latex]. In other words, the [latex]x[/latex]-coordinates of the intersection points are solutions of the equation [latex]x^2-2x-10=5[/latex]. Finding the intersection points graphically, as in figure 1, is equivalent to solving the equation [latex]f(x)=g(x)[/latex].

The intersection point(s) between the graphs of any two functions [latex]f(x)[/latex] and [latex]g(x)[/latex] can be found algebraically by setting the two functions equal to each other:

[latex]f(x) = g(x)[/latex]

At any intersection point [latex](x,y)[/latex], the value of [latex]x[/latex] is the same for both functions, as is the value of [latex]y[/latex]. In other words, [latex]f(x)=g(x)[/latex] means when the two functions have the same input [latex]x[/latex], and the output of the two functions [latex]f(x)[/latex] and [latex]g(x)[/latex] are equal (i.e., [latex]f(x)=g(x)[/latex]).

Example 1

Use Desmos to find the solution to the equation [latex]x^2-x-12=0[/latex].

Solution

Let [latex]f(x)=x^2-x-12[/latex] and [latex]g(x)=0[/latex]. Note that [latex]g(x)=0[/latex] is represented by the [latex]x[/latex]-axis, so we are finding where the graph of [latex]y=f(x)[/latex] intersects the [latex]x[/latex]-axis (i.e., the [latex]x[/latex]-intercepts).

Using Desmos we get:

From the graph, [latex]y=f(x)[/latex] intersects the [latex]x[/latex]-axis at (–3, 0) and (4, 0).

So the solution to the equation [latex]x^2-x-12=0[/latex] is [latex]x=–3,\;4[/latex].

 

Note. We can always check our solution by substituting the values into the equation:

When [latex]x=–3[/latex],

[latex]\begin{aligned}x^2-x-12&=0\\(-3)^2-(-3)-12&=0\\9+3-12&=0\\0&=0\;\;\;\;\text{True}\end{aligned}[/latex]

When [latex]x=4[/latex],

[latex]\begin{aligned}x^2-x-12&=0\\(4)^2-(4)-12&=0\\16-4-12&=0\\0&=0\;\;\;\;\text{True}\end{aligned}[/latex]

Example 2

Use Desmos to find the solution to the equation [latex]x^2-4=-x^2+14[/latex].

Solution

Let [latex]f(x)=x^2-4[/latex] and [latex]g(x)=-x^2+14[/latex]. then use Desmos to graph the two functions.

 

From the graph, the intersection points are (–3, 5) and (3, 5).

So, the solution of the equation [latex]x^2-4=-x^2+14[/latex] is [latex]x=–3,\;3[/latex].

Try It 1

Use Desmos to find the solution to the equation [latex]x^2-3=6[/latex].

Show Answer

[latex]x=-3,\;3[/latex]

The Relationship of Equivalent Equations, Solutions, Zeros, [latex]x[/latex]-Intercepts, and Points of Intersection

No matter the equation, [latex]f(x)=g(x)[/latex] can always be simplified to [latex]f(x)-g(x)=0[/latex]. The difference in two polynomial functions is always another polynomial function, so solving [latex]f(x)=g(x)[/latex] is equivalent to solving [latex]p(x)=0[/latex], where [latex]p(x)=f(x)-g(x)[/latex].

For example, the equation in Example 2, [latex]x^2-4=-x^2+14[/latex], can be simplified to [latex]2x^2-18=0[/latex]. [latex]x^2-4=-x^2+14[/latex] and [latex]2x^2-18=0[/latex] are equivalent equations so they have identical solutions. Consequently, solving [latex]x^2-4=-x^2+14[/latex] is equivalent to solving [latex]2x^2-18=0[/latex], which in turn is equivalent to finding the zeros of the function [latex]p(x)=2x^2-18[/latex]. In other words, solving [latex]x^2-4=-x^2+14[/latex] is equivalent to finding the intersection of [latex]p(x)=2x^2-18[/latex] with the [latex]x[/latex]-axis, which is where the [latex]x[/latex]-intercepts lie. Figure 2 shows this graphically.

Figure 2. Finding zeros of a function

The zeros of [latex]p(x)=2x^2-18[/latex] are [latex]x=-3,\;3[/latex].

The [latex]x[/latex]-intercepts of the graph of [latex]p(x)=2x^2-18[/latex] are (–3, 0) and (3, 0).

The graphs of [latex]f(x)=2x^2-18[/latex] and [latex]g(x)=0[/latex] intersect at [latex]x=-3,\;3[/latex].

The solutions of the polynomial equation [latex]2x^2-18=0[/latex] are [latex]x=-3,\;3[/latex].

Do you see the relationship between zeros, intersection points, [latex]x[/latex]-intercepts, and solutions?

Solving a Polynomial Equation

We now know what it means to solve a polynomial equation graphically, but what about solving the equation algebraically? Afterall, the intersection points of two curves will not always end up on integer values of [latex]x[/latex].

One way to solve a polynomial set equal to zero is to use the zero product property.

The Zero Product Property

Suppose we multiply two numbers together and get an answer of zero. What can you say about the two numbers? Could they be [latex]2[/latex] and [latex]5[/latex]? Could they be [latex]9[/latex] and [latex]–1[/latex]? No! The only way to get a product of zero, is to multiply by zero. So if a product equals zero, at least one of the factors must be zero. This idea is called the zero product property, and it is useful for solving polynomial equations that can be factored.

Using Factoring to Solve a Polynomial Equation

The zero product property makes it possible for us to solve factored polynomial equations. So, if the equation contains a polynomial that is not factored, the first thing we need to do is 1) simplify it so that it is equal to zero, and 2) factor.

The following video shows two more examples of using both factoring and the principle of zero products to solve a polynomial equation.

The next video shows that we can use previously learned methods to factor a trinomial in order to solve a polynomial equation.

What happens if we don’t have zero on one side of the equation? More often than not, we will start by simplifying the equation to get zero on one side.